The aim of the paper is to discuss the extreme points of subordination and weak subordination families of harmonic mappings. Several necessary conditions and sufficient conditions for harmonic mappings to be extreme points of the corresponding families are established.

MSC 2010: 30C45The universally prestarlike functions of order α ≤ 1 in the slit domain Λ = C [1;∞) have been recently introduced by S. Ruscheweyh. This notion generalizes the corresponding one for functions in the unit disk Δ (and other circular domains in C). In this paper, we obtain the coefficient inequalities and the Fekete-Szegö inequality for such functions.

In this paper, we obtain Fekete–Szegö inequalities for a generalized class of analytic functions $f\left(z\right)\in 𝒜$ for which $1+\frac{1}{b}\left(\frac{z{\left(D_{\alpha ,\beta ,\lambda ,\delta }^{n}f\left(z\right)\right)}^{\text{'}}}{D_{\alpha ,\beta ,\lambda ,\delta }^{n}f\left(z\right)}-1\right)$ ($\alpha ,\beta ,\lambda ,\delta \ge 0$; $\beta >\alpha$; $\lambda >\delta$; $b\in {ℂ}^{*}$; $n\in {\mathbb{N}}_0$; $z\in U$) lies in a region starlike with respect to $1$ and is symmetric with respect to the real axis.

The authors obtain the Fekete-Szegő inequality (according to parameters $s$ and $t$ in the region $s^2+st+t^2<3$, $s\ne t$ and $s+t\ne 2$, or in the region $s^2+st+t^2>3,$$s\ne t$ and $s+t\ne 2$) for certain normalized analytic functions $f\left(z\right)$ belonging to $k{\text{-UST}}_{\lambda ,\mu }^n(s,t,\gamma )$ which satisfy the condition $$\Re \left\{\frac{(s-t)z{\left(D_{\lambda ,\mu }^{n}f\left(z\right)\right)}^{\text{'}}}{D_{\lambda ,\mu }^{n}f\left(sz\right)-D_{\lambda ,\mu }^{n}f\left(tz\right)}\right\}>k\left|\frac{(s-t)z{\left(D_{\lambda ,\mu }^{n}f\left(z\right)\right)}^{\text{'}}}{D_{\lambda ,\mu }^{n}f\left(sz\right)-D_{\lambda ,\mu }^{n}f\left(tz\right)}-1\right|+\gamma ,z\in 𝒰.$$ Also certain...

MSC 2010: 30C45

In the present paper, a theorem for the starlikeness and convexity of multivalent functions involving certain inequalities is given. Some interesting consequences of the main result are also mentioned.